Variable Time Step Dynamics With Choice

Variable Time Step Dynamics With Choice Lev Kapitanski Department of Mathematics, University of Miami Coral Gables, FL 33124, USA and ˇ Sanja Gonzalez Zivanovi´ c Department of Mathematics and Computer Science, Barry University Miami Shores, FL 33161, USA

ABSTRACT

We develop a simple and general approach to study long term behavior of deterministic systems that switch regimes and have dwell times of variable length. We investigate the results of all possible as well as restricted, and/or controlled, switchings. To analyze all these situations, we introduce the notions of variable time step dynamics with choice and variable time step iterated function systems. We establish general sufficient conditions for the existence of global compact attractors of such systems and describe how they are related. Keywords Global Attractors, Dynamical Systems, Control, Dwell Time, and Iterated functions system. 1. INTRODUCTION We are interested in time evolution of systems that can and do switch their modes (regimes) of operation at discrete moments of time. The intervals between switching may, in general, vary. The number of modes (regimes) may be finite or infinite. Such systems are very common in life. In control theory the systems we talk about are often called hybrid, see [5, 6]. The switching times and the switching of the regimes may be deterministic or random. Here we will discuss the deterministic case only. A few examples may be helpful. The first example is a switched system, i.e., a special case of a hybrid system governed by a finite set of systems of ordinary differential equations, x˙ = fw(j) (x) on the interval [tj , tj+1 ),

(1)

where t0 < t1 < . . . are the switching times, and each fw(j) is taken from a finite set of

functions {f0 , . . . , fN −1 }. Here w(·) is a regime switching function, it maps the non-negative integers into the label set of the available regimes, {0, 1, . . . , N − 1}. Denote the states at the regime switching times by xj = x(tj ) and the time intervals between the switching (dwell time) by h(j) = tj+1 −tj . We can write the transition from h(j) xj to xj+1 symbolically as xj+1 = Sw(j) (xj ). Here h(j)

Sw(j) is a transformation that solves system (1), h(j)

i.e., for every y, Sw(j) (y) is the solution x(t) of the system x˙ = fw(j) (x) with the initial condition x(0) = y, evaluated at time t = h(j). The second example is a discrete version of (1), xj+1 = xj + h(j) fc(j) (xj ) ,

(2)

where h(j) is a variable, in general, time step. (In fact, (2) does not have to be related to (1) and could have a totally independent origin.) Again, we can write the transformation from the state xj h(j) to the next state, xj+1 , as xj+1 = Sw(j) (xj ), but h(j)

now Sw(j) is given explicitly by the right hand side of (2). In both examples, x takes on values in some region of a finite dimensional space Rd . However, it is not hard to come up with meaningful examples of infinite dimensional systems with continuous or discrete time where parameters switch during the evolution. The switching times and the regime switching function can be given in advance, or can be generated step-by-step depending on external or internal information. In the latter case, h(j + 1) and w(j + 1) may depend on the state xj , on the regime w(j), or even on xj−1 , w(j − 1), etc. The rule generating h(j+1) and w(j+1) can be viewed as a control law. In the present report we address the long term behavior of systems of the type just described. In

particular, we are interested whether or not a system possesses global attractor. A global attractor is a compact subset of the state space that attracts all bounded sets (and not just individual points). This is an important notion because in practice the states of the system are known only approximately. If the global attractor consists of one point, this point (state) is automatically asymptotically stable. The global attractors of nonlinear dissipative systems may have a very complicated geometry (as, e.g., the Lorenz attractor).

as a regime switching function. In a similar fashion, h ∈ ΣI can be viewed as h(0)h(1)h(2) . . . , and we call h a dwell time function, or a time step function. The sets ΣJ and ΣI are equipped with a shift operator, σ, that acts as follows: σ(w)(n) = w(n + 1). [In the language of symbolic dynamics, ΣJ is a full one-sided shift over the alphabet J , see [13].] Now, we view equation (3) as a “non-autonomous system” and apply the wellknown skew-product construction, [20], to obtain an “autonomous system.”

There is a large literature on asymptotic stability (of one state, the origin, x = 0) of both linear and nonlinear switched systems, see e.g. [17, 19] and references there. Outside of engineering applications, we should mention an interesting paper [18] on using switching controls in epidemic models. A number of papers are devoted to attractors of switched systems, mostly of the form (1) in the finite-dimensional setting, see e.g. [7, 8, 14].

Definition 1. The variable time step dynamics with choice on X associated with the maps Sjτ , j ∈ J , τ ∈ I, is the discrete time dynamics generated on the product space X = X × ΣJ × ΣI by the iterations of the map

In [11, 12, 22] we developed a simple and general approach to dynamics of fixed time step switching systems. We use the term dynamics with choice because we handle the systems more general than (1) and (2). In this paper we extend our approach to deal with variable time step. We give sufficient conditions for the existence of global compact attractors. One of those conditions may be hard to check in practice as we show with an example. 2. DYNAMICS WITH CHOICE Consider a set X (the state space) and a collection of maps Sjτ : X → X. The lower index, j, indicates the chosen regime. All available regimes are labeled by the points of some set J . The upper index, τ , is interpreted as the dwell time. The values of τ are taken from some interval I = [a, b], 0 < a ≤ b. Thus, Sjτ (x) is interpreted as the result of the regime j acting on the state x over the time period τ . Switching regimes and dwell times leads to the dynamics xn+1 = Sjτnn (xn ) .

(3)

We would like to be able to work with all the trajectories x0 , x1 , x2 , . . . generated this way with arbitrary jn ∈ J and τn ∈ I. One can think of several different mathematical formalizations of this dynamics. We will use two. Denote by ΣJ (respectively, ΣI ) the space of maps from the non-negative integers Z≥0 into J (respectively, I). Sometimes it is convenient to view a map, w ∈ ΣJ , as a one-sided infinite string (word) of symbols w(0), w(1), w(2), . . . from J ; we write w = w(0)w(1)w(2) . . . . Sometimes we refer to w

h(0)

S : (x, w, h) 7→ (Sw(0) (x), σ(w), σ(h)) .

(4)

The second formalization of the deterministic dynamics (3) is motivated by iterated function systems (IFS), [2, 3, 9]. Here one works not with points of X, but with its subsets. Define the maps F τ , τ ∈ I, acting on the subsets of X according to the rule [ F τ (A) = Sjτ (A) . (5) j∈J

Definition 2. The variable time step iterated function system on X associated with the maps Sjτ , j ∈ J , τ ∈ I, is the discrete time dynamics generated on the product space Y = 2X × ΣI by the iterations of the map F : (A, h) 7→ (F h(0) (A), σ(h)) .

(6)

This notion of a variable time step iterated function system seems to be new. In addition, we propose another useful variant of a variable time step iterated function system inside the state space. Definition 3. The variable time step iterated function system inside X associated with the maps Sjτ , j ∈ J , τ ∈ I, is the discrete time dynamics generated on 2X by the iterations of the map [ F : A 7→ F τ (A) . (7) τ ∈I

3. GLOBAL ATTRACTORS Consider a discrete time dynamics generated on a metric space Y by iterations of a map Φ : Y → Y . The global compact attractor of such semi-dynamical system is the smallest compact set A ⊂ Y that attracts all bounded sets. The

latter means that for every A ⊂ Y of finite diameter, and for any open neighborhood U of A, there exists N such that Φn (A) ⊂ U for all n ≥ N . Both definitions in the previous section present the variable time step dynamics (3) in the form of a discrete semi-dynamical system. The theory of attractors for continuous time as well as for discrete semi-dynamical systems is well developed, see e.g. [16, 21] and a brief account in [10]. We will apply this theory in our situation. To this end we first need to outfit the state space X and the maps Sjτ with certain properties. The following will be our basic assumptions. We assume that X is a complete metric space with metric dX , that the set J is a metric compact with metric dJ , and that each map Sjτ is continuous and bounded. Also, we assume that there is a measure of noncompactness, ψ, so that each map Sjτ is ψ-condensing. Recall that a function ψ : 2X → [0, +∞] is a measure of noncompactness on X iff (see [1]) (i) ψ(A) = 0 iff A is relatively compact; (ii) If A1 ⊂ A2 , then ψ(A1 ) ≤ ψ(A2 ) ; (iii) ψ(A1 ∪ A2 ) = max {ψ(A1 ), ψ(A2 )} ; (iv) There exists a constant c(ψ) > 0 such that

provided dJ (j1 , j2 ) dI (τ1 , τ2 ) ≤ δI .

≤

For example, the Kuratowski measure of noncompactness of a set A (denoted α(A)) is the infimum of the numbers  > 0 such that A admits a finite cover by sets of diameter less than . Recall that a map S : X → X is ψ-condensing (condensing with respect to ψ) iff ψ(S(A)) ≤ ψ(A) for any bounded A, and ψ(S(A)) < ψ(A) if ψ(A) > 0 (i.e., if A is not compact). The compact maps X → X are ψ-condensing. Also, in a Banach space X, any map of the form contraction + compact is ψ-condensing for any ψ. Now we introduce two additional assumptions. The first assumption is always true if the number of regimes (the set J ) is finite and all dwell times are the same. Assumption 1. For any closed, bounded set A ⊂ X, the maps Sjτ , restricted to A, depend uniformly continuously on j and τ . More precisely, given a closed, bounded A, for every  > 0 there exist δJ > 0 and δI > 0 such that sup dX (Sjτ11 (x), Sjτ22 (x)) ≤ 

x∈A

=

In the second assumption we use the following notation. Given a one-sided infinite string h ∈ ΣI , h[n] denotes the finite word made by the first n symbols in h, i.e., h[n] = h(0)h(1) · · · h(n − 1), and kh[n]k stands for h(0) + h(1) + · · · + h(n − 1). h[n] Also, we denote by Sw[n] the composition h[n]

h(n−1)

h(1)

h(0)

Sw[n] = Sw(n−1) ◦ · · · ◦ Sw(1) ◦ Sw(0) . Assumption 2. Assume there is a closed, bounded set B ⊂ X such that for every bounded A ⊂ X there exists T (A) > 0 such that h[n] Sw[n] (A) ⊂ B, for any word h ∈ ΣI such that kh[n]k > T (A) and any word w ∈ ΣJ . This assumption requiring the existence of an absorbing set is necessary for the existence of a global attractor. We are ready to state our main result, but first make a small adjustment to the definition of the variable time step iterated function system. We modify the map F τ in (5) by taking the closure of the union as follows [ Sjτ (A) . (8) F τ (A) =

|ψ(A1 ) − ψ(A2 )| ≤ c(ψ) dist(A1 , A2 ), where dist(A1 , A2 ) is the Hausdorff distance between A1 and A2 .

δJ and |τ1 − τ2 |

j∈J

Similarly, we adjust the map F in (7): F(A) =

[

F τ (A) .

(9)

τ ∈I

Also, in Definition 2, we replace the space 2X by the space of all closed nonempty subsets of X, which we denote by C(X). The Hausdorff distance furnishes a complete metric to this space, [15]. Theorem 4. Under the above assumptions on X, J , I, and Sjτ , we consider the corresponding variable time step dynamics with choice and iterated function system. 1. The variable time step dynamics with choice possesses a global compact attractor. This attractor, A, is a subset of the product space X ×ΣJ ×ΣI , and is invariant under the map S defined in (4), S(A) = A. 2. The variable time step iterated function system possesses a global compact attractor. This attractor, B, is a subset of the product space C(X) × ΣI , and is invariant under the map F defined in (6), F(B) = B.

3. The variable time step iterated function system inside X possesses a global compact attractor. This attractor, K, is a subset of C(X) and is invariant under the map F. 4. Define the set K ⊂ X as the union of all the points (which are closed subsets of X) of the attractor K. This is a compact set. The attractors A and B are related to the fractal K in the following sense: B = K × ΣI ,

A = K × ΣJ × ΣI . (10)

The proof of this theorem relies on the following lemma. Lemma 5. Under the above assumptions, the maps S, F, and F are condensing with respect to the appropriate measures of noncompactness on the spaces X × ΣJ × ΣI , C(X) × ΣI , and C(X) respectively. The measures of noncompactness in the lemma are defined as follows. If Y and Z are two complete metric spaces, with some measures of noncompactness ψY and ψZ , we construct a measure of noncompactness on the product space Y × Z as follows:

u : X × J × I → J × I that will determine the next regime and the next dwell time based on the current state, regime, and dwell time. Denoting the J and I components of u by uJ and uI , we have jn+1 = uJ (xn , jn , τn ) , (11) τn+1 = uI (xn , jn , τn ) . Note that the system of equations (3) and (11) defines a discrete time dynamics on the space X × J × I. Theorem 6. If the assumptions of Sec. 3 are satisfied, then for any continuous control map u, the system governed by equations (3) and (11) possesses a global compact attractor, M, in X × J × I. If (x, j, τ ) ∈ M, then, setting x0 = x, w(0) = j, and h(0) = τ , and defining recursively h(n)

xn+1 = Sw(n) (xn ), w(n + 1) = uJ (xn , w(n), h(n)), h(n + 1) = uI (xn , w(n), h(n)) , we obtain the infinite strings w ∈ ΣJ and h ∈ ΣI . It turns out that (x, w, h) belongs to the attractor A of the corresponding variable time step dynamics with choice. 5. EXAMPLE

ψY ×Z (A) = max{ψY (AY ), ψZ (AZ )}, where AY and AZ are the projections of A ⊂ Y × Z on the spaces Y and Z. On the space C(Y ) we define a measure of noncompactness as follows: ! [ ψC(Y ) (N ) = ψY A . A∈N

In this section we give an example of a twodimensional switched system that shows that, in practice, checking Assumption 2 of Sec. 3 may be quite tricky. We build the example by first patching a couple of systems of ODEs on the plane and then transplanting the result to an infinite cylinder. The first system is this:

Our approach allows us to easily handle dynamics with restricted choice in the variable time step setting. By restricted choice we mean that certain sequences of regime and/or dwell times switches may be forbidden. Mathematically this means that the full shift ΣJ may be replaced by a subshift (i.e., closed and shift-invariant subset of ΣJ ) ΛJ . Similarly, ΣI may be replaced by a subshift ΛI . The Definitions 1 and 2 with these replacements give rise to the variable time step dynamics with restricted choice and restricted iterated function systems. Theorem 4 does hold for such systems. 4. CONTROL IN DYNAMICS WITH CHOICE Within our general framework of Sec. 2 we see many opportunities to introduce controls. For example, depending on you goals, choose a map

x˙ = − y˙ =


y + x 1 − x2 − y 2 2 +y
x + y 1 − x2 − y 2 2 2 x +y x2

(12)

The origin is its unstable focus and x2 + y 2 = 1 is the stable limit-circle. In the neighborhood of the origin, we shall glue in two linear systems between which the switching will occur. Those systems are x˙ = x − 4y y˙ = x + y

(13)

x˙ = x − y y˙ = 4x + y,

(14)

where  is a positive parameter. This type of systems is well known in the stability theory of switched systems, [17]. For each (13) and (14), the origin is an unstable fixed point, but with the right switching between (13) and (14) (e.g., using (13) if xy > 0 and using (14) if xy < 0), the origin becomes globally asymptotically stable, see

[4]. Transfer systems (12), (13), and (14) to a cylinder by changing the coordinates x and y to s and θ, where x = es cos θ, y = es sin θ. In the (s, θ) coordinates system (12) reads s˙ = 1 − e2s ,

θ˙ = e−2s ,

(15)

system (13) reads s˙ =  −

3 sin(2θ), 2

1 θ˙ = (5 − 3 cos(2θ)), (16) 2

and system (14) reads s˙ =  +

3 sin(2θ), 2

1 θ˙ = (5 + 3 cos(2θ)). (17) 2

The half of the cylinder with s < 0 corresponds to the interior of the unit circle, and the half with s > 0 - to its exterior. We want system (15) to operate in the region s ≥ −1 and systems (16) and (17) to operate where s < −4. This will be achieved by adding the right sides of (15) with coefficient ζ(s) to the corresponding right sides of systems (16) or (17) with coefficient (1 − ζ(s)), where ζ(s) is a smooth, monotone increasing function such that ζ(s) = 0 for s ≤ −4 and ζ(s) = 1 for s ≥ −1. We write the resulting systems symbolically as   d s = fw(j) (s, θ) , (18) dt θ where w(j) = 1 corresponds to (16) combined with (15), and w(j) = 2 corresponds to (17). Note, that on the cylinder, system (15) and each of the two systems (18) have the circle s = 0 as the global compact attractor. However, depending on the strategy of switching between system number 1 and system number 2 of (18), the attractor may survive, or there may be no global attractor. To understand this we only need to understand how switching works for systems (13) and (14). Indeed, consider the corresponding switched system       d x x x = + Aw(j) (19) y y dt y where w(j) = 1 or 2 and    0 −4 0 A1 = , A2 = 1 0 4

−1 0

 (20)

We will exhibit an interval [a, b] ⊂ (0, +∞) such that for any choice of time steps τj from this interval and for any sequence of regime switching w(j), all trajectories starting not at the origin go to infinity. This will imply that on the cylinder, all trajectories of system (18) starting at the points with s < 0 (in fact, any disc in this region) will move in the direction of the circle s = 0.

The trajectories starting at the points with s > 0 will move to the same circle because in the xycoordinates s = 0 is the limit-circle of system (12). Thus, the variable time step system (18) with J = {1, 2} and I = [a, b] does have a global attractor. In the setting of Sec. 2, the global attractor of the variable time step dynamics with choice is A = {s = 0} × ΣJ × ΣI and the fractal K is the circle s = 0. Choosing the wrong interval for the time steps will destroy this picture. Some trajectories of system (19) will converge to the origin. On the cylinder this means s → −∞, and there is no global attractor for (18). We will give an example of a bad interval. To find a “good” interval, it suffices to find those τ for which the solutions     x0 x0 eτ eτ A1 and eτ eτ A2 y0 y0 of (13) and (14) are both farther from the origin than the initial condition [x0 , y0 ]T by some factor. In other words,     x0 x0 τ τ A1,2 e ke k≥γk k (21) y0 y0 for some γ > 1 and all [x0 , y0 ]T on the unit circle. Just for an example of some “good” interval, we find that     1 x0 x0 k2 keτ A1,2 k2 ≥ k y0 y0 2 when 5/9 < cos(4τ ) ≤ 1, and take those τ , which in addition guarantee that e2τ /2 ≥ γ 2 > 1, for some γ > 1. If  = 0.3, we may take τ ∈ I = [1.326, 1.810]. A different argument is needed to find the “bad” intervals. Note, that the eigenvalues of the matrices eτ Aj are e±2τ i , and the trajectories corresponding to sequences without switching do not go to the origin. However, both products eτ A1 eτ A2 and eτ A2 eτ A1 have real, negative eigenvalues, and the largest eigenvalue, λ(τ ), is such that |λ(τ )| < 1, provided τ satisfies | cos(2τ )| < 3/5. Thus, given an  > 0, we find intervals of τ such that e2τ |λ(τ )| < δ < 1, for some δ. Those intervals are bad. Denote M1τ = e2τ eτ A1 eτ A2 and M2τ = e2τ eτ A2 eτ A1 . We can choose any sequence τj from any of the bad intervals and look τj at the trajectories [xj+1 , yj+1 ]T = Mw(j) [xj , yj ]T , for any infinite string w of 1’s and 2’s. The distance from the origin to the consecutive images of any circle (x0 )2 + (y0 )2 = r2 will go to 0 exponentially fast (although some points in the images

will go to infinity). If  = 0.3, and I = [0.5, 1.06], the dynamics with choice does not have a global compact attractor. 6. CONCLUSION To investigate the long term behavior of deterministic systems that switch regimes at discrete moments of time and have dwell times of variable length, and to investigate the results of all possible as well as restricted, and/or controlled, switchings, we have introduced the notions of a variable time step dynamics with choice and a variable time step iterated function system. Using these notions as a rigorous foundation, we establish general sufficient conditions for the existence of global compact attractors of such systems and describe how they are related. Unlike in the case of systems without switching, one of the conditions (Assumption 2) may be hard to check in practice. We give a two-dimensional example that illustrates this difficulty. 7. REFERENCES [1] R. R. Akhmerov, M. I. Kamenski˘ı, A. S. Potapov, A. E. Rodkina, B. N. Sadovski˘ı, Measures of noncompactness and condensing operators, Operator Theory: Advances and Applications, 55. Birkhuser Verlag, Basel, 1992. [2] M. F. Barnsley, Fractals everywhere, Second edition. Academic Press Professional, Boston, MA, 1993. [3] M. F. Barnsley, Superfractals, Cambridge University Press, Cambridge, 2006. [4] M. Branicky, “Stability of switched and hybrid systems”, Proc. of the 33rd Conference on Design and Control, Lake Buena Vista, FL, 1994, pp. 3498-3503. [5] M. Branicky, Studies in Hybrid Systems: Modeling, Analysis, and Control, Sc.D. thesis, MIT, 1995 [6] M. Branicky, Introduction to hybrid systems, in D. Hristu-Varsakelis and W.S. Levine (eds.), Handbook of Networked and Embedded Control Systems, pp. 91-116. Birkhauser, 2005. [7] D. N. Cheban, “Compact Global Attractors of Control Systems”, Journal of Dynamical and Control Systems, vol.16 (2010), no.1, pp. 23–44. [8] D. N. Cheban, Global Attractors of SetValued Dynamical and Control Systems. Nova Science Publishers Inc, New York, 2010. [9] J. E. Hutchinson, “Fractals and selfsimilarity”, Indiana Univ. Math. J. 30, 1981, no. 5, pp. 713–747.

[10] L. Kapitanski, I. N. Kostin, “Attractors of nonlinear evolution equations and their approximations”, (Russian) Algebra i Analiz 2, 1990, no. 1, pp. 114–140; translation in Leningrad Math. J. 2, 1991, no. 1, pp. 97– 117 ˇ [11] L. Kapitanski, S. Zivanovi´ c,: “Dynamics with choice”, Nonlinearity 22, 2009, pp. 163– 186. ˇ [12] L. Kapitanski, ; S. Zivanovi´ c, : “Dynamics with a range of choice”, Reliable Computing, vol. 15, no. 4, 201, pp. 290-299. [13] B. P. Kitchens, Symbolic dynamics. Onesided, two-sided and countable state Markov shifts, Universitext: Springer-Verlag, Berlin, 1998. [14] P. E. Kloeden, “Nonautonomous attractors of switching systems”, Dynamical Systems, vol. 21, no. 2, 2006, pp. 209–230. [15] K. Kuratowski, Topology, Volume I. New York - London - Warszawa, Academic Press: Polish Scientific Publishers, 1966. [16] O. A. Ladyzhenskaya, Attractors for semigroups and evolution equations, Lezioni Lincee. [Lincei Lectures] Cambridge University Press, Cambridge, 1991. [17] D. Liberzon, Switching in systems and control, Systems & Control: Foundations & Applications, Birkh¨auser Boston, Inc., Boston, MA, 2003. [18] X. Liu, P. Stechlinski, “Pulse and constant control schemes for epidemic models with seasonality”, Nonlinear Analysis: Real World Applications, 12, 2011, pp. 931–946. [19] M. Margaliot, “Stability analysis of switched systems using variational principles: an introduction”, Automatica J. IFAC, 42, 2006, no. 12, pp. 2059–2077. [20] G. R. Sell, “Nonautonomous differential equations and topological dynamics”, I, II, Trans. Amer. Math. Soc., 127, 1967, pp. 241–262, pp. 263–283. [21] G. R. Sell, Y. You, Dynamics of evolutionary equations, Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002. ˇ [22] S. Zivanovi´ c, Attractors in Dynamics with Choice, PhD Thesis, University of Miami, Coral Gables, Fl, USA, 2009.